Zena M. Ariola, Jan Willem Klop, and Detlef Plump. Bisimilarity in term graph rewriting. Inform. and Comput., 156(1-2):2--24, 2000. ISSN 0890-5401.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....oe ; r oe ] t , where C contains n 1 holes. Define the unmarked substitution oe by xoe = e(xoe ) and let C = e(C ) Then e(s ) C[loe; loe] R C[roe; roe] e(t ) in n steps. 1 The origin of the notion bisimilarity is explained in [AKP99] 2 The set Tw (F ; V ) of well marked terms over F and V is the subset of T (F ; V ) such that t 2 Tw (F ; V ) if and only if, for every pair t 1 ; t 2 of subterms of t , mark(root(t 1 ) mark(root(t 2 ) implies t 1 = t 2 . For example, the ....

....rewrite relation ) nc R Tw (F ; V ) Theta Tw (F ; V ) w.r.t. R is defined by: s ) nc R t if s ;R t by a marked rule l r such that every marked variable in r appears also in l . Both the graph rewrite relation as defined in [Plu93, KKSV94, AK96, AKP99, Plu98] and the noncopying rewrite relation as defined in [KO95] can be viewed as a special case of marked rewriting. In contrast to graph rewriting, the definition of noncopying rewriting does not specify how the right hand side r should be marked. It is only necessary that a reduct of a ....

Z. M. Ariola, J. W. Klop, and D. Plump. Bisimilarity in term graph rewriting. Information and Computation, 1999. To appear.

.... k ) rank(s 1 ) If we restrict ourselves to the model in which variables are not marked at all (which means that variables are maximally shared) then the definition of graph rewriting used here becomes equivalent to the corresponding definitions in [BEG 87, Plu93a, Plu93b, KKSV94, AK96, AKP98] Example 3.6 Let R = feq(x; x) trueg as in Example 2.1. If variables are unmarked, then eq 0 (eq 1 (0 2 ; 0 3 ) eq 1 (0 2 ; 0 3 ) does not reduce to eq 0 (true 4 ; true 4 ) because 0 2 and 0 3 are not shared and thus the rule eq 1 (x; x) true 4 cannot be ....

.... and complete implementation (in what sense) of term rewriting ffl Under which conditions is )R terminating ffl Under which conditions is )R confluent The discussion of these problems is beyond the scope of this note, we refer to [BEG 87, Plu93a, Plu93b, KKSV94, KO95, AK96, Ohl97, AKP98] instead. Acknowledgements: I thank Masahito Kurihara for a valuable comment on a previous version of this note. ....

Z. M. Ariola, J. W. Klop, and D. Plump. Bisimilarity in Term Graph Rewriting. 1998. Submitted. 2

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Zena M. Ariola, Jan Willem Klop, and Detlef Plump. Bisimilarity in term graph rewriting. Information and Computation, 156(1/2):2-24, 2000. 11

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Zena M. Ariola, Jan Willem Klop, and Detlef Plump. Bisimilarity in term graph rewriting. Information and Computation, 156(1/2):2-24, 2000.

....rewriting is presented. Results and counterexamples are given for plain applications of rewrite rules, extensions with the operations of collapsing and copying, and with both operations together. Collapsing and copying together constitute bisimilarity of term graphs. These results are extended in [AKP98] by considering term graph rewriting modulo bisimilarity. This notion of rewriting turns out to be a proper extension of term rewriting but behaves with respect to termination and confluence exactly like term rewriting. Computations by term graph rewriting terminate more often than computations ....

Zena M. Ariola, Jan Willem Klop, and Detlef Plump. Bisimilarity in term graph rewriting. Report SEN-R9801, CWI, Amsterdam, 1998.

....term and term graph rewriting. The paper focusses on soundness of term graph rewriting with respect to term rewriting, on completeness for proving validity of equations and for computing term normal forms, on termination and confluence, and on term graph narrowing. The contents of the paper [AKP99] have been described in the First Annual Progress Report: In [CG98] the authors extend their previous work [CG97] on the categorical description of possibly cyclic term graph rewriting using suitable 2 categories to the rewriting of possibly cyclic term graphs. They show that this presentation ....

Zena M. Ariola, Jan Willem Klop, and Detlef Plump. Bisimilarity in term graph rewriting. Information and Computation, 1999. To appear.

....it is straightforward to show that every sequence of term rewrite steps can be simulated if both collapsing and copying are present. 25 g f 0 0 0 g f 0 0 ) g 0 f f 0 OE g 0 f f 0 0 Figure 13: Simulation of a term rewrite step Theorem 5. 2 (Completeness of ) bi and ) [7, 6]) For all term graphs G and H, the following are equivalent: 1) term(G) term(H) 2) G ) bi H. 3) G] H] Proof By the definitions of ) bi and ) it is clear that (2) implies (3) and (3) implies (1) by soundness of ) So it remains to show that (1) implies (2) By Lemma 5.1, for ....

....by ) coll . 28 We conclude this subsection by considering graph reducubility by ) copy . The result below follows from the fact that if all term rewrite rules are left linear, then for every term rewrite derivation t u there is a term graph rewrite derivation Mt ) copy Mu (see [6]) Theorem 5.9 Every left linear term rewriting system is strongly graph reducible by ) copy . 5.3 Bibliographic notes Completeness of ) coll for proving equational validity was shown in [88] Graphreducibility was first considered in [15] where the lifting of certain term rewrite strategies ....

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Zena M. Ariola, Jan Willem Klop, and Detlef Plump. Bisimilarity in term graph rewriting. Information and Computation. To appear.

....on term graph narrowing. The rst part of the habilitation thesis [Plu99a] also surveys term graph rewriting, while the second part studies termination and con uence the key properties for term and term graph rewriting in the setting of general graph transformation. The contents of the paper [AKP00] was described in the First Progress 34 Report. Term graphs play a central role in the functorial presentation of multialgebras, recently proposed by Corradini and Gadducci. Multi algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from ....

Zena M. Ariola, Jan Willem Klop, and Detlef Plump. Bisimilarity in term graph rewriting. Information and Computation, 156(1/2):2-24, 2000.

....every sequence of term rewrite steps can be simulated if both collapsing and copying are present. 26 CHAPTER 1. TERM GRAPH REWRITING g f 0 0 0 g f 0 0 ) g 0 f f 0 OE g 0 f f 0 0 Figure 1.13: Simulation of a term rewrite step Theorem 1.5. 2 (Completeness of ) bi and ) [7,6]) For all term graphs G and H , the following are equivalent: 1) term(G) term(H) 2) G ) bi H . 3) G] H ] Proof By the definitions of ) bi and ) it is clear that (2) implies (3) and (3) implies (1) by soundness of ) So it remains to show that (1) implies (2) By Lemma 1.5.1, ....

....) coll . ut 1.6. TERMINATION 29 We conclude this subsection by considering graph reducubility by ) copy . The result below follows from the fact that if all term rewrite rules are left linear, then for every term rewrite derivation t u there is a term graph rewrite derivation Mt ) copy Mu (see [6]) Theorem 1.5.9 Every left linear term rewriting system is strongly graph reducible by ) copy . ut 1.5.3 Bibliographic Notes Completeness of ) coll for proving equational validity was shown in [90] Graphreducibility was first considered in [15] where the lifting of certain term rewrite ....

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Zena M. Ariola, Jan Willem Klop, and Detlef Plump. Bisimilarity in term graph rewriting. Information and Computation. To appear.

....not hold in general. To test term graph rewriting for confluence, there is a method based on the analysis of critical pairs of term graph rewrite steps [Plu94] A survey of (positive and negative) confluence results for term graph rewriting over various classes of equation sets is given in [AKP98] Modularity. It is known that the union of two equation sets E and E 0 , which are both terminating under term rewriting and have disjoint function symbols, may yield a nonterminating set E [ E 0 . In contrast, termination of E and E 0 under term graph rewriting carries over to E [ E 0 ....

Zena M. Ariola, Jan Willem Klop, and Detlef Plump. Bisimilarity in term graph rewriting. Information and Computation, 1998. To appear.

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Zena M. Ariola, Jan Willem Klop, and Detlef Plump. Bisimilarity in term graph rewriting. Inform. and Comput., 156(1-2):2--24, 2000. ISSN 0890-5401.

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